FOLIATIONS INVARIANT BY RATIONAL MAPS 1

FOLIATIONS INVARIANT BY RATIONAL MAPS ´ CHARLES FAVRE AND JORGE VITORIO PEREIRA

Abstract. We give a classification of pairs (F , φ) where F is a holomorphic foliation on a projective surface and φ is a non-invertible dominant rational map preserving F .

1. Introduction Our goal is to give a classification of pairs (F, φ) where F is a holomorphic singular foliation on a projective surface X, and φ : X 99K X is a dominant rational map preserving F. When φ is birational, the situation is completely understood, see [5, 15]. When φ is not invertible, to the best of our knowledge the only result available in the literature is due to M. Dabija and M. Jonsson [6], and concerns the classification of pencil of curves in P2 preserved by holomorphic self-maps of the projective space. Our approach to this problem is based on foliated Mori theory [2, 3, 12, 13] which address the problem of classification of holomorphic foliations on surfaces by describing the numerical properties of their cotangent bundle in all birational models. Our main result gives a classification of pairs (F, φ) as above up to birational conjugacy and ramified cover. Theorem A. Suppose F is a holomorphic singular foliation on a projective surface X without rational first integral, and φ : X 99K X is a dominant non-invertible rational map preserving F. Then up to a birational conjugacy, there exists a φinvariant Zariski open dense subset U ⊂ X such that U is a quotient of C2 by a discrete subgroup Γ ⊂ Aff(C2 ) acting properly on C2 and: • φ lifts to an affine map on C2 ; • F is defined in C2 either by dx or by d(y +ex ) in suitable affine coordinates. Note that this implies F is non singular and defined by a closed 1-form on the open set U, and has a Liouvillian first integral. We shall call any rational map φ which lifts to an affine map in C2 like in the theorem Lattès-like. This class already appears in the classification of endomorphisms of Pk , k ≥ 1 having a non-trivial centralizer by Dinh and Sibony [8]. Let us comment briefly on which surfaces U may appear. When Γ is a subgroup of (C2 , +) of rank r ∈ {1, 2, 3, 4}, then U is either a torus (r = 4); a locally trivial fibration over C∗ with fiber some elliptic curve (r = 3); C×E for some elliptic curve, or C∗ ×C∗ (r = 2); or C×C∗ (r = 1). In particular any projective1 compactification Date: July 7, 2009. 2000 Mathematics Subject Classification. Primary: 37F75, Secondary: 14E05, 32S65. The first author was partially supported by the project ECOS-Sud No. C07E01, and the second by CNPq-Brazil. 1Otherwise more examples arise like non-k¨ ahlerian Kodaira surfaces. 1

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´ CHARLES FAVRE AND JORGE VITORIO PEREIRA

of U is birational to a torus, to P1 × E for some elliptic curve E, or to P1 × P1 . Note that each of these surfaces admit Lattès-like maps. In general, when Γ is no longer a subgroup of (C2 , +), one obtains finite quotients of these examples. In fact, there are strong restrictions for a Lattès-like map on a surface X to preserve a foliation in C2 which induces a holomorphic singular φ-invariant foliation on X. Our results are actually more precise, and we give normal forms for both the foliation and the map, see Theorems 4.3, 4.4 below in the case F has no first integral and Theorem 4.6 when F is tangent to a pencil of curves. As a consequence of the classification, we obtain the following result which extends [5, Corollaire 1.3] to non-invertible maps. Corollary B. Suppose φ : X 99K X is a dominant non-invertible rational surface map which preserves a holomorphic foliation F which is not tangent to a rational or an elliptic fibration. Then φ preserves at least two foliations. The assumption is necessary, since the map (x, y) 7→ (x2 , xy 2 ) on C2 does not preserve any other foliation than {x = cst}. Our approach also gives an alternative to the delicate analysis of the reduced fibers done by Dabija-Jonsson in [6]. It allows us to extend their results to arbitrary foliations. Note that the following result is not an immediate consequence of the previous results since it gives a classification up to PGL(3, C). Theorem C. Suppose φ : P2 → P2 is a holomorphic map of degree d ≥ 2 preserving a foliation F. Then deg(F) = 0 or 1, and in appropriate homogeneous coordinates [x : y : z] on P2 , one of the following cases holds: (1) F is the pencil of lines given by d(x/y), and φ = [P (x, y) : Q(x, y) : R(x, y, t)] with P, Q, R homogeneous polynomials of degree d; (2) F is induced by d log(xλ yz −1−λ ) with λ ∈ C \ Q, and φ = [xd : y d : z d ]; (3) F is induced by log(xξ yz −1−ξ ) with ξ a primitive 3-rd root of the unity and φ = [z d : xd : y d ]; (4) F is induced by d(xp y q /z p+q ) with p, q ∈ N∗ , p 6= q, gcd{p, q} = 1, and φ = Ql [xd : y d : R(x, y, z)] with R = z δ i=1 (z p+q + ci xp y q ) with d = δ + l(p + q), c i ∈ C∗ . (5) F is induced by the 1-form d log(xyz −2 ), φ = [y d : xd : R(x, y, z)] with Ql R = z δ i=1 (z 2 + ci xy) and d = δ + 2l, ci ∈ C∗ . In [10], we extend our classification to rational maps preserving webs, inspired by the work of Dabija-Jonsson [7] on endomorphisms preserving families of lines. The plan of the paper is as follows. In Section 2, we recall basic facts about foliations and their singularities. We then describe how the cotangent bundle of a foliation behaves under the action of a rational map in Proposition 2.1. From this key computation, we deduce a simple proof of Theorem C. In Section 3, we present the basics of foliated Mori theory, following [2]. We deduce from it two general statements (Proposition 3.4 and 3.5) that are important intermediate results. Section 4 contains the classification up to finite covering and birational conjugacy of invariant foliations without first integral of Kodaira dimension 0 and 1 (Theorem 4.3 and 4.4), and of invariant fibrations (Theorem 4.6). The proofs of Theorem A and Corollary B are then given at the end of the paper. Acknowledgements. We thank Romain Dujardin for his comments on a preliminary version of this paper.

FOLIATIONS INVARIANT BY RATIONAL MAPS

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2. Transformation of the cotangent bundle In this section by a surface X we mean a smooth compact complex surface. We denote by T X its tangent bundle, by Ω1X its sheaf of holomorphic 1-forms and by KX := Ω2X its canonical line bundle. 2.1. Basics on foliations. A (singular holomorphic) foliation F on a surface X is determined by a section with isolated zeroes v ∈ H 0 (X, T X ⊗ T ∗ F) for some line bundle T ∗ F on X called the cotangent bundle of F. Two sections v, v 0 define the same foliation if and only if they differ by a global nowhere vanishing holomorphic function. The zero locus of the section v is a finite set Sing(F) called the singular locus of F. A point x ∈ / Sing(F) is said to be regular. Concretely, given an open contractible Stein cover {Ui } of X, F is described in each Ui by some vector field vi with isolated zeroes such that vi = gij vj on Ui ∩ Uj for some non-vanishing holomorphic functions gij ∈ O∗ (Ui ∩ Uj ). The cocycle {gij } determines the cotangent bundle of F, and the collection {vi } induces a global section v ∈ H 0 (X, T X ⊗ T ∗ F). Integral curves of the vector fields {vi } patch together to form the leaves of F. Outside Sing(F), a local section of T ∗ F is given by a holomorphic 1-form along the leaves. A foliation F can be described in a dual way by a section with isolated singularities ω ∈ H 0 (X, Ω1X ⊗ N F) with isolated zeroes for some line bundle N F called the normal bundle of F. In a contractible Stein open cover, ω is determined by holomorphic 1-forms ωi . The tangent spaces of leaves are then the kernels of 1-forms ωi . Notice that the collection of 1-forms {ωi } determine the same foliation as the collection of vector fields {vi } if and only if ωi (vi ) ≡ 0 for all i. At a regular point of F, one can contract a local section of T F with a holomorphic 2-form yielding a local section of N ∗ F. We thus have the important isomorphism: (2.1)

KX = N ∗ F ⊗ T ∗ F

Suppose φ : Y → X is a dominant holomorphic map between two surfaces X, Y , and F is a foliation on X determined by a collection of 1-forms {ωi } on some open cover {Ui } of X. The holomorphic 1-form φ∗ ωi on φ−1 (Ui ) may have in general non-isolated zeroes. We denote by ω ˆ i any holomorphic 1-form on φ−1 (Ui ) with isolated zeroes which is proportional to φ∗ ωi . The collection {ˆ ωi } then defines a holomorphic foliation that we call the pull-back of F by φ and denote by φ∗ F. A singular point x for a foliation F is said to be reduced if the foliation is locally defined by a vector field v whose linear part is not nilpotent, and such that the quotient of its two eigenvalues is not a positive rational number (it may be zero or infinite). Reduced singularities satisfy the following property: for any composition of point blow-ups π above p, then the invertible sheaf T ∗ (π ∗ F)−π ∗ T ∗ F is determined by an effective π-exceptional divisor. A separatrix at x is a germ of curve C passing through x such that C \ {x} is a leaf of F. The singularity of F at x is said to be dicritical if there exist infinitely many separatrices at this point. A reduced singularity is not dicritical. A reduced foliation on a surface is a foliation with all its singularities reduced. If F is a reduced foliation on X, and φ : Y → X is a composition of point blow-ups then φ∗ F is again reduced. It is a theorem of Seidenberg that for any foliation F there exists a composition of point blow-ups φ : Y → X such that φ∗ F is reduced.


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2.2. Pull-back of foliations. Suppose π : X 0 → X is a composition of point blow-ups and F 0 is a foliation on X 0 . Then we can push-forward F 0 outside the exceptional components of π. This defines a foliation on the complement of a finite set in X. By Hartog’s theorem, this foliation extends to X in a unique way: we denote it by π∗ F 0 . If φ : Y 99K X is a dominant meromorphic map and F is a foliation on X then φ∗ F is defined as follows. Let Γ be a desingularization of the graph of φ. Thus there are two holomorphic maps π : Γ → Y and f : Γ → X such that π is a composition of point blow-ups and f = φ◦π as as indicated in the diagram below. By definition, φ∗ F is the foliation π∗ f ∗ F on X. Γ? ~~ ???f ~ ?? ~~ ? ~ ~ φ X _ _ _ _ _ _ _/ Y π

In the sequel, a foliated surface (X, F) is a smooth compact complex surface endowed with a singular holomorphic foliation. A map φ : (Y, G) 99K (X, F) of foliated surfaces is a dominant meromorphic map φ : Y 99K X such that G = φ∗ F. Let us introduce a few more notation. We let Ind(φ) be the set of indeterminacy points of φ. In terms of the diagram above, it is the finite set of points p ∈ X such that f π −1 (p) has positive dimension. For any divisor D in X, we set φ∗ D := π∗ f ∗ D. It is a linear map which preserves effectivity. It induces maps between the Picard and Neron-Severi groups of X and Y that we again denote by φ∗ : Pic(X) → Pic(Y ), and φ∗ : NS(X) → NS(Y ). Denote by ∆φ the ramification divisor of φ. It is locally defined by the vanishing of det Dφ, where Dφ denotes the differential of φ. Its support is the critical set of φ, and it satisfies the equation (2.2)

KX = φ∗ KY + ∆φ .

The following result is our key technical tool. Proposition 2.1. Suppose φ : (Y, G) 99K (X, F) is a dominant meromorphic map between smooth foliated surfaces. Then one has φ∗ T ∗ F = T ∗ G − D in Pic(Y ) for some divisor whose support is included in the critical set of φ and satisfies D ≤ ∆φ . Pick any critical component E, and assume that φ(E) is a reduced singularity of F when it is a point. (1) If E is generically transverse to G then ordE (D) = ordE (∆φ ) > 0; (2) If E is G-invariant, then ordE (D) ≥ 0. Proof of Proposition 2.1. Suppose F is given by a collection of holomorphic 1-forms {ωi } on an open Stein cover {Ui } of X. On the open set φ−1 (Ui ) \ Ind(φ), we may write φ∗ ωi = hi · ω ˆ i with ω ˆ i a holomorphic 1-form with isolated zeroes and hi a holomorphic function whose zero set is included in the critical set of φ. The divisors div(hi ) patch together and yields a global effective divisor D0 . Outside Sing(F), ωi is a local generator for the invertible sheaf N ∗ F over Ui , and the same is true with ω ˆ i for N ∗ G over φ−1 (Ui ) \ Sing(G). Whence φ∗ N ∗ F = N ∗ G − D0 . By (2.1) and (2.2), we get φ∗ T ∗ F = T ∗ G + D0 − ∆φ . This proves the first part of the proposition with D = ∆φ − D0 .


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For the second part, we pick an irreducible component E of the critical set of φ. By our assumption we know that either E is not contracted, or it is contracted to a reduced singularity of F. Suppose first E is transversal to G. Then it cannot be contracted to a point since a reduced singularity admits only finitely many separatrices. At a generic point on E, and in suitable coordinates we may write φ(x, y) = (xa , y). On the other hand, we may suppose that F is determined by the form dy, hence φ∗ (dy) = dy. We conclude that D0 = 0, and ordE (D) = ordE (∆φ ). Suppose next E is G-invariant but is not contracted. Again we may assume φ(x, y) = (xa , y), but then F is determined by dx. Now φ∗ (dx) = axa−1 dx and det Dφ = axa−1 so that D0 = (a − 1)E = ∆φ , and ordE (D) = 0. Finally suppose E is G-invariant and contracted to a point p which is a reduced ˆ → X be a composition of point blow-ups such that the map singularity. Let π : X ˆ ˆ φ : Y 99K X satisfying π ◦ φˆ = φ does not contract E. Write Fˆ := π ∗ F. As p is a reduced singularity, T ∗ Fˆ = π ∗ T ∗ F + E 0 with E 0 an effective divisor supported on the exceptional set of π. As E is not contracted by φˆ we may apply our previous ˆ for some divisor D ˆ such that arguments. We may thus write φˆ∗ T ∗ Fˆ − T ∗ G = −D ˆ ordE (D) = 0. Now we get ˆ − φˆ∗ E 0 φ∗ T ∗ F − T ∗ G = φˆ∗ π ∗ T ∗ F − T ∗ G = φˆ∗ T ∗ Fˆ − T ∗ G − φˆ∗ E 0 = −D ˆ + ordE (φˆ∗ E 0 ) = ordE (φˆ∗ E 0 ) ≥ 0. We conclude that ordE (D) = ordE (D)

2.3. Proof of Theorem C. We actually prove a stronger result, see the proposition below. Recall that deg(F), the degree of a foliation F on P2 , is defined as the number of tangencies between F and a generic line `. If F is defined by a section with isolated zeroes ω ∈ H 0 (P2 , Ω1P2 ⊗ N F) and N F = OP2 (k) for some k, then the restriction ω|` is a section of H 0 (P1 , Ω1P1 ⊗ OP1 (k))). By definition the number of zeroes of ω|` is equal to deg(F) so that N F = OP2 (deg(F) + 2)

and

T ∗ F = OP2 (deg(F) − 1).

Recall that the algebraic degree of a rational map φ : P2 99K P2 is by definition the degree of φ−1 ` for a generic line `. If we apply Proposition 2.1 to reduced foliations on P2 then we obtain the following result. Proposition 2.2. Let φ : (P2 , F) 99K (P2 , F) be a dominant rational map of algebraic degree d > 1. Suppose F is reduced or φ does not contract any curve. Then deg(F) ≤ 1. Proof. One has deg(φ∗ T ∗ F) = d·(deg(F)−1). Our assumption and Proposition 2.1 implies D = T ∗ F − φ∗ T ∗ F is effective. Thus (deg(F) − 1) − d · (deg(F) − 1) ≥ 0, i.e. deg(F) ≤ 1. Back to the proof of Theorem C. Suppose φ is holomorphic. It cannot contract curves, hence by the previous proposition, deg(F) = 0 or 1. Having degree 0 means F is given by the fibration {x/y = cst}, whichshows we are in case (1). When or (b) d log(xλ yz −1−λ ) with F has degree 1, it is defined by (a) d log xy exp yz λ ∈ C∗ . We now note that critical components of φ are necessarily F-invariant, as follows from Proposition 2.1.

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If we are in case (a) then there are only two φ-invariant algebraic curves, the lines {xy = 0}, and their union is totally invariant by φ. Hence φ or its square is equal to [xd : y d : R(x, y, z)]. A simple computation shows that a holomorphic map of this form cannot leave a foliation in the type (a) above. Suppose now we are in case (b). If λ ∈ / Q then F has only three invariant algebraic curves which are totally invariant by φ. We are thus in cases (2) and (3) of p q y = cst} the theorem. If λ = p/q, then the foliation is given by the fibration { xzp+q which admits a unique reducible component {xy = 0}. This component is necessarily totally invariant, hence φ = [xd : y d : R] or [y d : xd : R] with R homogeneous of degree d. In the former case, since {xy = 0} is a totally invariant fiber, we may find a polynomial P of degree d such that Rp+q (x, y, 1) = (xp y q )d P ( xp1yq ). This implies all irreducible factors of R(x, y, 1) are of the form 1 + cxp y q with c ∈ C∗ as required. In the latter case, the equation becomes Rp+q (x, y, 1) = (xq y p )d P ( xp1yq ). Since R(x, y, 1) is a polynomial, this forces p = q = 1. 3. Foliated Mori theory 3.1. Basics. For the convenience of the reader, we recall the main aspects of foliated Mori theory as developed by Miyaoka, McQuillan [12], Brunella [2, 3] and Mendes [13]. Its goal is to classify holomorphic foliations on projective surfaces in terms of the intersection properties of their cotangent bundles. The first important result of this theory is due to Miyaoka. Theorem 3.1 (Miyaoka). If (X, F) is a reduced foliation on a projective surface and T ∗ F is not pseudo-effective, then F is tangent to a rational fibration. We may thus turn our attention to reduced foliations (X, F) with T ∗ F pseudoeffective. The next fundamental result is due to McQuillan. Theorem 3.2 (McQuillan). Suppose (X, F) is a reduced foliation and T ∗ F is pseudo-effective. Then one can find a regular birational map µ : X → X0 to a projective surface X0 with at most cyclic quotient singularities such that F0 := µ∗ F is smooth at any of the singularities of X0 and T ∗ F0 is a Q-line bundle which is nef. Recall that a germ of foliation defined near a quotient singularity C2 /G with G a finite subgroup of GL(2, C) is smooth if it is the image of a germ of smooth foliation in C2 by the quotient map. For sake of convenience, we introduce the following terminology. Definition 3.3. A nef foliation is a foliation F on a projective surface X with at most cyclic quotient singularities such that: F is smooth at any of the singular points of X; it is reduced at all regular points of X; and T ∗ F is a nef Q-line bundle. The preceding two theorems can be thus rephrased as follows: either F is tangent to a rational pencil, or it is nef in some birational model. The classification of nef foliations is done according to the values of two invariants called the Kodaira dimension and the numerical Kodaira dimension. The Kodaira dimension of a reduced foliation (X, F) is by definition the Kodaira-Iitaka dimension of its cotangent bundle, that is: kod(F) = lim sup n→∞

log h0 (X, T ∗ F ⊗n ) . log n


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It is not hard to check that two reduced foliations that are birationally conjugated have the same Kodaira dimension. The numerical Kodaira dimension of a reduced foliation (X, F) is defined in terms of the Zariski decomposition of T ∗ F. Write T ∗ F = PF + NF in NSQ (X) with PF a nef Q-divisor and NF an effective Q-divisor with contractible support such that PF · NF = 0. Such a decomposition is unique. Then we set:  −∞ when T ∗ F is not pseudo-effective;    0 when PF = 0; ν(F) = 1 when PF2 = 0 but PF 6= 0;    2 when PF2 > 0. Again two birationally conjugated reduced foliations have the same numerical Kodaira dimension. One can now summarize the classification of foliated surfaces into the following table: ν(F) −∞ 0

kod(F) −∞ 0

1 1

−∞ 1

2

2

Description Rational fibration F is the quotient of a foliation generated by a global holomorphic vector field by a finite cyclic group. Hilbert Modular foliation Riccati foliation Turbulent foliation Nonisotrivial elliptic fibration Isotrivial fibration of genus ≥ 2 General type

Recall that a foliation is a Riccati (resp. turbulent) foliation, if there exists a fibration π : X → B whose generic fiber is rational (resp. elliptic), and transversal to F. A foliation (X, F) is a Hilbert modular foliation if, up to birational morphisms, there exists a Zariski open subset U of X which is isomorphic to the quotient space H2 /Γ where H is the upper half plane and Γ is an irreducible lattice in PSL(2, R)2 and the restriction of F to U is the quotient of the horizontal fibration of H2 by Γ. 3.2. General properties of φ-invariant nef foliations. Proposition 3.4. Suppose (X, F) is a nef foliation, and φ is a dominant noninvertible rational map preserving F. Then, we have: (T ∗ F)2 = 0 and φ∗ T ∗ F = T ∗ F in Pic(X) . Proof. Denote by e(φ) the topological degree of φ: by assumption it is an integer greater or equal to 2. By Proposition 2.1, we have the equality of invertible sheaves φ∗ T ∗ F = T ∗ F − D in Pic(X) with D effective (apply the proposition first to the minimal desingularization of X and project the equality to down to X). Because T ∗ F is nef, we get (T ∗ F)2 ≥ (φ∗ T ∗ F)2 . But the latter term is always greater of equal to e(φ) · (T ∗ F)2 , see for instance [9, Corollary 3.4], hence (T ∗ F)2 = 0. We also get 0 ≤ φ∗ T ∗ F · T ∗ F ≤ (T ∗ F)2 = 0, hence φ∗ T ∗ F = λT ∗ F in NSR (X) for some positive constant λ by Hodge index theorem. Since φ preserves the lattice in NSR (X) generated by classes associated to curves, and since φ∗ T ∗ F ≤ T ∗ F, we have λ = 1. We thus have φ∗ T ∗ F = T ∗ F in NS(X) whence D = 0 numerically. But D is effective so that D = 0 in Pic(X). This concludes the proof.

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Proposition 3.5. Suppose (X, F) is a nef foliation, and φ is a dominant noninvertible rational map preserving F. Denote by C the closure of the F-invariant compact curves, and write U := X \ C. Then either U is empty and F is tangent to a fibration; or U is a Zariski-dense open subset of X. In the latter case, we may contract finitely many curves on X such that F remains a nef foliation and φ induces a holomorphic proper unramified finite covering φ : U → U. In particular, X \ U is a φ-totally invariant proper analytic subset of X which is maximal for the inclusion. Note U may equal X. Notice also that U may have quotient singularities. When it is the case φ is unramified in the orbifold sense, which means that the critical locus of φ is empty outside the singular points, and φ has finite fibers and is proper. For sake of convenience, we introduce the following terminology Definition 3.6. Suppose (X, F) is a nef foliation, and φ is a dominant rational map preserving F. Then the foliation F is φ-prepared if the complement U of all F-invariant curves is a non-empty dense Zariski open subset of X which is φinvariant, and such that the restriction map φ : U → U is an unramified orbifold cover. The previous proposition says that any nef foliation without a first integral admits a model in which it is φ-prepared. Proof. By Jouanolou’s theorem, if F admits infinitely many invariant compact curves, then it is tangent to a fibration. Thus U is either empty or the complement of finitely many algebraic curves, hence Zariski-dense. For the rest of the proof, assume we are in the latter case. We let Γ be a desingularization of the graph of φ, and µ : Γ → X, f : Γ → X be the two natural projections with µ birational, and f = φ ◦ µ. Proposition 3.4 implies φ∗ T ∗ F = T ∗ F. Thus there is no critical curve intersecting U according to Proposition 2.1 (1). It is clear that for an arbitrary point p ∈ U \ Ind(φ) its image φ(p) is still in U. Let now p ∈ Ind(φ) ∩ U, and pick a neighborhood V around p. Then f induces a holomorphic map from the neighborhood µ−1 (V ) of the divisor C := µ−1 (p) in Γ to a neighborhood of f (C). Since C is contractible, the intersection form on the free abelian group of divisors supported on C is negative definite. For any divisor E supported on f (C), one has E 2 = e−1 (f ∗ E)2 < 0 with e the topological degree of f restricted to V . Whence f (C) is contractible. We may thus contract the divisor f (C) to a point q. In doing so the map f becomes holomorphic at p and sends it to q. Since p is a quotient singularity, q is also a quotient singularity. Moreover, because T ∗ F is nef, any component of f (C) is orthogonal to T ∗ F, hence F induces a nef foliation on the new surface, and q is a cyclic quotient singularity. Note that this procedure does not create new indeterminacy point in U. We may thus repeat this procedure finitely many times, and arrive at a situation where φ is holomorphic on U. It remains to show that U is invariant by φ. Since F has no dicritical singularities, for any point of indeterminacy p ∈ X the divisor f (µ−1 (p)) is F-invariant, hence included in C. Suppose by contradiction that there exists a connected curve C ⊂ C contracted by φ to a point p ∈ U. By the preceding remark, C does not intersect Ind(φ). Replace C by the connected component of φ−1 (p) containing it, and pick a small tubular neighborhood U of C. Then φ induces a holomorphic map from


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U onto its image, whose critical set is included in C. We thus get an unramified finite covering from U \ C onto the complement of p in a small neighborhood. As before we can contract C, and after finitely iterations get a new model in which µ(f −1 (U)) ⊂ U. The inclusion is in fact an equality since f is surjective from Γ onto X. We conclude that φ induces a proper finite unramified map from U onto itself. 4. The classification 4.1. Kodaira dimension 0. If (X, F) is a nef foliation of Kodaira dimension zero then, according to [12, Theorem IV.3.6], (X, F) is the quotient of a foliation (Y, G) with trivial canonical bundle on a smooth surface Y by a finite cyclic group H acting on Y without pseudo reflections. Thus the action has isolated fixed points and the points in X below these fixed points are singular. We summarize the possibilities in the following table, for the order of the groups see [16]. Ambient Space Y

Foliation G

Sesquielliptic surface Abelian Surface

Isotrivial elliptic fibration any vector field not tangent to an elliptic fibration Suspension of a representation π1 (E) → C∗ Suspension of a representation π1 (E) → C any irrational vector field in the Lie algebra of C∗ × C∗ any vector field in the Lie algebra of C∗ × C with non trivial projections to both factors

extension of an elliptic by C∗ compactified as a P1 -bundle extension of an elliptic by C compactified as a P1 -bundle C∗ × C∗ compactified as P2 or P1 × P1 C∗ × C compactified as P2 or P1 × P1

order of H 1,2,3,4,6 1,2,3,4,5, 6,8,10,12 1,2,3,4,6 1,2 1,2,3,4,6 1,2

From the classification, we deduce the following useful fact. Fact: when the original foliation has no first integral, then the space Y is a compactification of a complex Lie group G with abelian Lie algebra and the foliation G, when restricted to the Lie group, is induced by a Lie subalgebra. In all these cases G is the quotient of (C2 , +) by a subgroup Γ of translations, the pullback of G to C2 is a linear foliation and H is generated by a cyclic element h ∈ GL(2, C) with both eigenvalues distinct from one. Lemma 4.1. Let φ : (X, F) 99K (X, F) be a rational self-map of a nef foliation with kod(F) = 0. Assume F is φ-prepared. If we write (X, F) as the quotient of (Y, G) by a cyclic group H as in the beginning of this section then the map φ lifts to a rational map φˆ : (Y, G) 99K (Y, G). Remark 4.2. Below we give a topological proof. Another proof can be obtained in the spirit of [5, p.209-210]. Proof. Let U be the complement of all F-invariant curves. Because π : Y → X is a finite map and U is Zariski dense, it suffices to lift the restriction of φ to U. Note that the preimage of U by π coincides with the Lie group G alluded to above. By assumption, the map φ|U is an orbifold covering of U. Since the restriction to G = π −1 (U) of the natural quotient map π : Y → X is also an orbifold covering ,

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the composition map g = φ ◦ π : G → U is an orbifold covering . It follows from the diagram ˆ|G ? ∃φ

G @_ _ _/ G @@ g @@ π @ π φ|U @ /U U that to lift φ|U to φˆ : Y 99K Y it suffices to check that g∗ π1 (G) ⊂ π∗ π1 (G), where g∗ : π1 (G) → π1orb (U) is the natural map induced by g. For the definition of the orbifold fundamental group π1orb and its basic properties, the reader can consult [11, 17]. Since g is an orbifold covering , g∗ is injective. In particular, every element in g∗ π1 (G) distinct from the identity has infinite order. To conclude it is sufficient to prove that any element f 6= id ∈ π1orb (U) of infinite order belongs to π∗ π1 (G). The group π1orb (U) can be interpreted as a subgroup of Aff(2, C). Write f (z) = Az + B with A ∈ GL(2, C) and B ∈ C2 . Recall from the fact above that π1orb (U) is an extension of a finite cyclic group generated by some element h ∈ GL(2, C) which is not a pseudo-reflection by a subgroup of translations canonically isomorphic to π1 (G). If A 6= id, then A is a power of h, hence is periodic P of period k > 1. Since k−1 i ◦k k h is not a pseudo-reflection, one has f (z) = A z + · B = z . This i=0 A contradicts our assumption. Whence f is a translation, and belongs to π∗ π1 (G). This concludes the proof. Theorem 4.3. Let (X, F) be a foliation on a projective surface with kod(F) = 0 and without first integral. Suppose φ is a dominant non-invertible rational map preserving F. Then up to birational conjugacy and to a finite cyclic covering of order N , we are in one of the following five cases. κ0 (1): The surface is a torus X = C2 /Λ, F is a linear foliation and φ is a linear diagonalizable map. In this case N ∈ {1, 2, 3, 4, 5, 6, 8, 10, 12}. κ0 (2): The surface is a ruled surface over an elliptic curve π : X → E whose monodromy is given by a representation ρ : π1 (E) → (C∗ , ×). In affine coordinates x on C∗ and y on the universal covering of E, F is induced by the 1-form ω = dy + λx−1 dx where λ ∈ C, and φ(x, y) = (xk , ky) where k ∈ Z \ {−1, 0, 1}. In this case N ∈ {1, 2, 3, 4, 6}. κ0 (3): Same as in case κ0 (2) with ρ : π1 (E) → (C, +), ω = dy + λdx where λ ∈ C, φ(x, y) = (ζx + b, ζy) where b ∈ C, and ζ ∈ C∗ with |ζ| = 6 1. In this case N ∈ {1, 2}. κ0 (4): The surface is P1 × P1 , the foliation F is given in affine coordinates by the form λx−1 dx+dy and φ(x, y) = (xk , ky) with λ ∈ C∗ and k ∈ Z\{−1, 0, 1}. In this case N ∈ {1, 2}. κ0 (5): The surface is P1 × P1 , the foliation F is given in affine coordinates by the form λx−1dx + µy −1 dy and φ(x, y) = (xa y b , xc y d ) where λ, µ ∈ C∗ , with M = ac db ∈ GL(2, Z), with λ/µ ∈ / Q+ , |ad−bc| ≥ 2, and M diagonalizable over C. In this case N ∈ {1, 2, 3, 4, 6}.


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Proof. Most of the proof follows immediately from the classification stated in the beginning of this section and Lemma 4.1. What is perhaps not completely evident is the assertion that φ is a linear diagonalizable map in cases κ0 (1) and κ0 (5). Let us comment on that. Suppose we are in case κ0 (1), and φ is not diagonalizable. Then in C2 we can write φ(x, y) = (ζx + y, ζy) for some ζ ∈ C∗ , and φ preserves a lattice Λ ⊂ C2 . Forgetting about the complex structure, we may now view φ as a linear map on R4 preserving Z4 with eigenvalues ζ and ζ¯ (each of multiplicity 2). The characteristic polynomial of φ is thus the square of a quadratic polynomial X 2 − aX + b with integral coefficients. From the Jordan decomposition of φ, we see that there exists a unique φ-invariant 2 dimensional real plane P . In R4 , it is given by P = ker (φ2 − aφ + bid), and is hence defined over Z. In C2 , it equals P = {y = 0} and is invariant under complex conjugation. We have thus proved that the intersection of Λ with the complex line {y = 0} is a rank 2 lattice. It follows that the foliation dy has a compact leaf. Since translations act transitively on C2 /Λ by preserving the foliation, we see that all leaves of the foliations are elliptic curves which implies the foliation to have a first integral. A contradiction. In the case κ0 (5), the matrix is again diagonalizable. Otherwise, one has φ(x, y) = (xd , xy d ) with d ∈ Z, and the invariant foliation is the rational fibration given by {x = cst}.

4.2. Kodaira dimension 1. The next result classifies the pairs (F, φ) when F has Kodaira dimension one and does not admit a rational first integral. Theorem 4.4. Let (X, F) be a foliation on a projective surface with kod(F) = 1 and without first integral. Suppose φ is a dominant non-invertible rational map preserving F. Then up to birational conjugacy and to a finite cyclic covering generated by an automorphism τ of order N , we are in one of the following two cases. κ1 (1): The surface X is P1 × P1 , the foliation is a Riccati foliation given in coordinates by the form ω=

m dx dy + + µxn dx , y (k − 1) x

and φ(x, y) = (λx, xm y k ), where λ, µ ∈ C∗ , k ∈ Z \ {−1, 0, 1}, m ∈ Z, n ∈ Z \ {−1}, such that λ is not a root of unity and k = λn+1 . Moreover n∈ / {−2, −1, 0} if m = 0. In this case, N ∈ {1, 2}, and τ = (±x±1 , ±y ±1 ). κ1 (2): The surface X is P1 × E where E is an elliptic curve, the foliation is a turbulent foliation induced by the form ω = dy + xn dx , where x is a coordinate on P1 and y on the universal covering of E, and φ(x, y) = (λx, λn+1 y) with λ ∈ C∗ not a root of unity, n ∈ Z \ {−2, −1, 0}. In this case, τ = (ζx, ξy), N = 1 when n ≥ 1, and N ∈ {1, 2, 3, 4, 6} otherwise. Proof. As before, we may suppose (X, F) is a nef foliation such that the complement of the compact F-invariant curves is Zariski dense and totally invariant by φ. Under the additional condition that kod(F) = 1, we shall need the following result.

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Lemma 4.5. There exists a rational or elliptic fibration π : X → P1 whose generic fiber is transversal to F and that admits one or two F-invariant fibers. Moreover, one can find f ∈ Aut(P1 ) which is not periodic, such that π ◦ φ = f ◦ π. In particular, the F-invariant fibers are totally invariant by φ. Finally singularities of the ambient space X are all located on F-invariant fibers. A proof is given at the end of this section. 1. Suppose first that π is a rational fibration. Then we may make a sequence of modifications centered on one of the F-invariant fibers such that X becomes P1 × P1 . Note that the condition on F being nef may not be satisfied anymore, but we can keep the properties listed in Lemma 4.5. Let us consider the component ∆⊥ of the critical set of φ which is not invariant by the fibration. A non-invertible rational map of P1 has at least two critical points, hence ∆⊥ · π −1 (b) ≥ 2. On the other hand, by Propositions 2.1 and 3.5, ∆⊥ is an F-invariant compact curve that is totally invariant by φ, so that ∆⊥ · π −1 (b) = 2. If ∆⊥ is irreducible, the fiber product ∆⊥ ×π X is a two-fold covering of X ramified over ∆⊥ . Replacing X by this fiber product if necessary, we may assume that ∆⊥ has two irreducible components C1 and C2 . We now pick coordinates (x, y) ∈ P1 × P1 such that π(x, y) = x, C1 = {y = 0}, C2 = {y = ∞}. We assume π −1 (∞) is an F-invariant fiber, and the other Finvariant (if it exists) is π −1 (0). The foliation F can thus be defined by a form ω = y −1 dy + h(x)dx for some rational function h. Poles of h correspond to Finvariant fibers, hence we may write h(x) = xn p(x) for some n ∈ Z, and some polynomial p ∈ C[x] with p(0) 6= 0. We also have φ(x, y) = (f (x), a(x)y k ) with a(x) ∈ C(x) and f (x) = x + 1 or = λx with λ ∈ C∗ not a root of unity. Assume first that f (x) = λx. Since F is φ-invariant, we have φ∗ ω = b · ω for some meromorphic function b. Expanding this equation in terms of x and y, and identifying both hand sides, we find that xn (kp(x) − λn+1 p(λx)) = a0 (x)/a(x). The left-hand side is holomorphic outside 0 ∈ C, thus a(x) = a0 xm for some m ∈ Z. 1/k−1 By changing the coordinate y by a0 y, we may assume a0 = 1. Therefore, we obtain xn+1 kp(x) − λn+1 p(λx) = m . Elementary considerations now show that we necessarily have m dx dy + + µxn dx . φ(x, y) = (λx, xm y k ), and ω = y (k − 1)x with λ ∈ C∗ , µ ∈ C, k ≥ 2, m ∈ Z, n ∈ Z \ {−1}, such that λ is not a root of unity and k = λn+1 . Since F has no first integral, µ is non zero. One can then compute T ∗ F by relating it to π ∗ KP1 , see for instance [2, §4]: T ∗ F = π ∗ OP1 (−2) + max{1, −n}π −1 (0) + max{1, n + 2}π −1 (∞) if m 6= 0, (4.1) T ∗ F = π ∗ OP1 (−2) + max{0, −n}π −1 (0) + max{0, n + 2}π −1 (∞) if m = 0 . Whence kod(F) = 1 if and only if m 6= and n 6= −1; or m = 0 and n 6= −2, −1, 0 as required. Assume now that f (x) = x + 1. The fiber π −1 (0) being not φ-invariant, it is not F-invariant, and thus h(x) is a polynomial. Similarly to the previous case, we need to solve kh(x) − h(x + 1) = a0 (x)/a(x). Because the left-hand side has no poles a(x) must be constant, and consequently h must satisfies the functional equation


13

kh(x) − h(x + 1) = 0, which is impossible. This concludes the proof of the theorem in the case π is a rational fibration. 2. Suppose now that π is an elliptic fibration. By Lemma 4.5, the action of φ on the base of the fibration is not periodic, hence the fibration is isotrivial. As in the previous case, pick coordinates x ∈ P1 such that the fiber π −1 (∞) is F-invariant, and the other possible F-invariant fiber is π −1 (0). Then f (x) = λx with λ ∈ C∗ not a root of unity, or f (x) = x + 1. We also pick an affine coordinate y on the universal covering of E. Suppose first f (x) = λx. After a base change of the form x 7→ xl , one can assume that X = P1 × E for some elliptic curve E. We may thus write φ(x, y) = (λx, ϕ(y)) for some non-invertible ϕ ∈ End(E), and the foliation F is induced by a form ω = dy + h(x)dx with h ∈ C(x) with poles only at 0 and ∞. Denote by ϕ0 the multiplier of ϕ. It is an integer if E has no complex multiplication, and a quadratic integer otherwise. Since |ϕ0 |2 is the topological degree of f , we have |ϕ0 | > 1. The φ-invariance of F reads in this case as λh(λx) = ϕ0 · h(x). Thus h(x) = µxn , ϕ0 = λn+1 with n ∈ Z \ {−1} and µ ∈ C∗ . Replacing x by µ1/n+1 x, we may take µ = 1. This shows φ = (λx, λn+1 y) and ω = dy + xn dx. Finally T ∗ F can be computed as before. One finds that it satisfies (4.1). Hence kod(F) = 1 if and only if n 6= −2, −1, 0 as required. Suppose now f (x) = x + 1. Then there is only one F-invariant fiber, namely π −1 (∞). Since C is simply connected, the monodromy of the fibration is trivial, and we may assume X = P1 × E, φ(x, y) = (x + 1, ϕ(y)) and F is induced by a oneform ω = dy + h(x)dx as before. The φ-invariance of F implies h(x + 1) = ϕ0 · h(x). But this equation has no solution when |ϕ0 | > 1. This concludes the proof. Proof of Lemma 4.5. Since kod(F) = 1 and F does not admit a rational first integral, for suitably large n, the natural map X → P(H 0 (T ∗ F ⊗n ))∗ induced by the complete linear series induces a fibration π : X → B whose generic fiber is either rational or elliptic. By construction, the class of a generic fiber of π is a multiple of (T ∗ F). The foliation is moreover transversal to a generic fiber of π, see [2, §9.2]. Since φ∗ T ∗ F = T ∗ F in Pic(X) by Proposition 3.4, the map φ preserves this fibration and induces a holomorphic map f on the base that is invertible. Suppose first that f = id in B. Then φ induces a holomorphic map on a generic fiber whose topological degree is e(φ) ≥ 2. In particular, replacing φ by a suitable iterate, we may assume that it admits at least three repelling fixed points. Let C be the irreducible component of the fixed point set of φ passing through such a point p. Then we have π(C) = B, and C · π −1 (b) ≥ 3 for any b ∈ B. At any point near p, the differential of φ has two eigenvectors, one with eigenvalue 1 and tangent to C, and the other one with eigenvalue of modulus > 1 and tangent to the fiber. Since φ preserves F and the foliation is transversal to π, we conclude that C is an F-invariant curve. By Proposition 3.5, the set of F-invariant compact curves is totally invariant by φ. But no non-invertible holomorphic self-map on a rational or an elliptic curve admits a totally invariant finite set of cardinality greater or equal to 3. We conclude that f 6= id. The same argument shows that f is not periodic, and thus the base B is either elliptic or rational. Note that we may push this argument further. Indeed, if f admits no totally invariant finite subsets, then the fibration has no F-invariant fiber. In this case, F is a suspension and has Kodaira dimension 0. Therefore B is a rational curve, and

14


f has either one or two totally invariant points, each one of them corresponding to an F-invariant fiber of the fibration. To conclude the proof, we need to show that the singular set of X is included in the F-invariant fibers. To do so, pick an arbitrary point p ∈ X not lying on an F-invariant fiber. A local neighborhood U of p is given by the quotient of C2 by a cyclic group generated by a map h(x, y) = (ζx, ξy) for some roots of unity ζ and ξ. Denote by g the natural projection g : C2 → U . The foliation g ∗ F is smooth since F is a nef foliation. The fibration induced by π ◦ g in C2 has to be transversal to g ∗ F otherwise F and the fibration would have some tangencies near p. Using the invariance by h, we conclude that π ◦ g(x, y) = x, and F is given by dy. Now at a generic point of the fiber containing p, the differential of the map π has rank 1, because f is invertible. This forces ζ = 1, whence X is smooth at p. 4.3. The case of a fibration. We now deal with foliations admitting rational first integrals. Theorem 4.6. Let φ be a dominant non-invertible rational map preserving a fibration π : X → B. Up to birational conjugacy and to a finite cyclic cover, we are in one of the following four mutually exclusive cases. (Fib1) The surface X = B ×F is a product of two Riemann surfaces with g(F ) ≥ 1 and g(B) ≤ 1, the fibration is the projection onto the first factor B×F → B, and φ = (f (x), ϕ(y)) is a product map with f not periodic. (Fib2) The surface is the torus X = E × E for some elliptic curve, and φ is skew product φ = (f (x), ϕ(x)(y)) with f not periodic, and ϕ : E → End(E) is holomorphic and non constant. (Fib3) The fibration is elliptic and the action of φ is periodic on the base. In other words, some iterate of φ is an endomorphism of an elliptic curve over the function field C(B). (Fib4) The fibration is rational, π : X := P1 × B → B, and φ is a skew product φ(x, y) = (f (x), ϕ(x)(y)) where f is a holomorphic map on B, and ϕ is rational map from B to the space of rational maps of a fixed degree on P1 . Proof. Let f be the induced map on the base. We may assume π is not a rational fibration, otherwise we are in case (Fib4). If f is periodic, since φ is not invertible the generic fiber is elliptic and we are in case (Fib3). If f is not periodic, the fibration is isotrivial: there exists a Riemann surface F such that π −1 (b) is isomorphic to F for almost all b. By the semi-stable reduction theorem, see [1, III.10], up to birational conjugacy, a local model near a singular fiber is given by a quotient D × F by a cyclic group generated by a map of the form (x, y) 7→ (ζx, h(y)) with ζ a root of unity, and h a finite order automorphism of F . To any such singular the order of its associated group. fiber π −1 (b) with b ∈PB, we let n(b) ∈ N∗ beP The divisors DB = (n(b) − 1)[b] and D = (n(b) − 1)π −1 (b) induce natural orbifold structures on B and X respectively and we denote by B orb , X orb the associated orbifolds. Then one can rephrase the semi-stable reduction theorem by saying that π : X orb → B orb is a locally trivial fibration in the orbifold sense. Any rational map preserving such a fibration is holomorphic, see for instance [4]. Since f : B orb → B orb is holomorphic and not periodic, we are in one of the following situations: (B, DB ) = (P1 , 0), (P1 , p), (P1 , p + q), (P1 , 2p), or (E, 0) for some elliptic curve (here p 6= q). For monodromy reasons, the cases (P1 , p) and (P1 , 2p) are excluded. A base change of order 2 allows one to reduce the case (P1 , p + q) to


15

(P1 , 0). In the latter case, we have X = P1 × B. Since by assumption g(B) ≥ 1, the map φ is a product: φ(x, y) = (f (x), ϕ(y)). In the case B is an elliptic curve, since we assume X to be k¨ ahlerian, it is a torus up to a finite cover. By Poincaré irreducibility theorem, X is a product of two elliptic curves B × B 0 . When the curves are not isogeneous, we fall into case (Fib1). Otherwise, we fall into either case (Fib1) or (Fib2) of the theorem. This concludes the proof. 4.4. Modular foliations. Let us briefly recall the geometric context in which modular foliations arise. We refer to [14] for a detailed description of the construction. Let Γ be an irreducible lattice in PSL(2, R)2 . The quotient X = H2 /Γ is a quasiprojective variety which may be projective or not. If not it admits a projective ¯ by adding finitely many cusps. When X is not projective X ˆ compactification X ¯ will stand for the minimal desingularization of the cusps of X. To keep the notation ˆ will be set equal to X when X itself is already projective. We point out uniform X ˆ that X is not necessarily smooth, we are resolving only the cusps, but has at worst cyclic quotient singularities. The two natural projections H2 induce two foliations by holomorphic disks on ¯ and X. ˆ In X, ˆ the bidisk. We denote by F and G their respective images in X, X ∗ ∗ ∗ ∗ T F and T G are nef and satisfy T F ⊗ T G = KXˆ ⊗ OXˆ (D) with D supported ˆ → X. ¯ Although the cotangent on the exceptional divisor of the projection µ : X ¯ are not locally free, we can still write T ∗ F + T ∗ G = KX¯ sheaves of F and G in X ¯ if we interpret T ∗ F, T ∗ G ∈ NSR (X) ¯ as the direct images under µ of in NSR (X) ˆ T ∗ F, T ∗ G ∈ NSR (X). It is a fact that the two classes T ∗ F and T ∗ G are not proportional, see for instance [14, §2.2.4]. Proposition 4.7. Suppose F is a modular foliation. Then any rational map φ preserving F is invertible. Proof. We give two independent arguments. ˆ that preserves F in the notation First argument. We may assume φ is a map on X ¯ above. For sake of convenience, we shall also write φ for its induced map on X. ∗ −1 Since T F is nef, we may apply Proposition 3.5. Here U coincides with µ (X), ¯ is hence φ is an unramified holomorphic finite covering from X to itself. Since X ¯ normal, φ is holomorphic on X also whose critical set is empty. We conclude that ¯ we have φ∗ KX¯ = KX¯ and φ∗ T ∗ F = T ∗ F, which implies in the space NSQ (X), ∗ ∗ ∗ φ T G = T G. Now the two-dimensional vector space generated by T ∗ F and T ∗ G is φ∗ -invariant and contains a nef class ω of positive self-intersection. Since φ∗ is holomorphic, we get e(φ)ω 2 = (φ∗ ω)2 = ω 2 , hence φ is an automorphism. Second argument. By Proposition 3.5, φ is unramified covering X, hence preserves S, the set of singular points of X. We deduce that φ induces a non-ramified finite covering from X \ φ−1 (S) onto X \ S. It can thus be lifted to a map φˆ between the universal covering s of these spaces which are both isomorphic H2 minus a discrete set. By Hartog’s theorem, φˆ extends as a map from H2 to itself without ramification points. In explicit coordinates (x, y) ∈ H2 where the lifting of F is given by the projection the first factor we can thus write b y) = a1 x + b1 , a2 (x)y + b2 (x) φ(x, c1 x + d1 c2 (x)y + d2 (x) where a1 , b1 , c1 , d1 are constants, a2 , b2 , c2 , d2 are holomorphic functions in H, both satisfying ai di − bi ci = 1. Because φb descends to H2 /Γ it must satisfy φb Γ φb−1 ⊂


16

Γ. We point out that Γ is a Zariski dense subgroup of G, see for instance [14, proof of Theorem 1]. Taking Zariski closures in G we see that φb G φb−1 ⊂ G. The specialization of this last equation to certain elements of G allows one to conclude that the functions a2 , b2 , c2 and d2 are indeed constants. Thus φb ∈ G and φb Γ φb−1 = Γ. Consequently φX is a biholomorphism (indeed an isometry with respect to the products of Poincaré metrics). 4.5. Proof of Theorems A and Corollary B. Let us begin with Theorem A. Pick F a foliation without first integral. By Miyaoka’s and McQuillan’s theorems, we may assume X is a model (with at most quotient singularities) in which F has reduced singularities and T ∗ F is nef. By Proposition 3.4, (T ∗ F)2 = 0 hence F can not be of general type. Thanks to the classification of foliated surfaces, we are in one of the following three cases: either F is a modular foliation, or kod(F) = 0, or kod(F) = 1. The modular case is excluded by Proposition 4.7. When kod(F) = 0, we may apply Theorem 4.3. It is clear that Theorem A holds in this case: the lift of F to C2 is a foliation induced by a constant vector field. When kod(F) = 1, we apply Theorem 4.4. When F is a Riccati foliation, then if we consider the map ϕ : C2 → C∗ × C∗ given by mx x , exp y − (x, y) 7→ exp n+1 (n + 1)(k − 1) then ∗

ϕ

dy m dx µex n + + µx dx = dy + dx , y k−1 x n+1

which is induced by a vector field. On the other hand, ϕ−1 ◦φ◦ϕ = (x+µ, ky+lx+ν) for some µ, ν and l and is thus Lattès-like. When F is a turbulent foliation, one x , y). Then considers the map ϕ : C × E → C∗ × E given by ϕ(x, y) := (exp n+1 x e ∗ −1 µ µ n+1 ϕ ω = dy + n+1 dx and ϕ ◦ φ ◦ ϕ = (x + µ, e y) with e = λ . This concludes the proof of Theorem A. Proof of Corollary B. Let φ : X 99K X be a dominant rational map preserving a foliation F. Suppose kod(F) = 0 and F has no first integral. Then there exists ˆ a dominant map fˆ : X ˆ 99K X, ˆ a foliation F, ˆ and a cyclic a projective surface X, ˆ ˆ ˆ ˆ group of automorphisms G ⊂ Aut (X) such that (φ, X, F) is in the list given by Theorem 4.3, G preserves Fˆ and commutes with φˆ and (φ, X, F) is birationally ˆ conjugated to the quotient situation on X/G. In cases κ0 (2), κ0 (3) and κ0 (4), the map φˆ preserves a unique rational fibration. This fibration is automatically G-invariant, hence φ preserves a rational pencil. In the case κ0 (1), the lift M of φˆ to C2 is diagonalizable. Since M G = GM and G is cyclic, M and G have two fixed points in common on P(C2 ). The projective space P(C2 ) naturally parameterizes the set of all linear foliations, hence φ preserves the two associated foliations. In the case κ0 (5), the same argument applies. When kod(F) = 1 and F has no first integral, we apply Theorem 4.4. The arguments are essentially the same. In the case κ1 (1), the group G that may appear are explicit, and the rational fibrations given by dx and y −1 dy + m[(k − 1)x]−1 dx are both G-invariant. In the case κ1 (2) there is a unique invariant elliptic fibration. Finally when F is a fibration which is neither rational nor elliptic, we fall into case (Fib1).


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